C.a.R. > Applications > Triangle Geometry > Nine point circle

The blue circle is the circle through the midpoints of the sides A* of the green triangle. It is also the circle through the bases of the heights F, and the circle through the midpoints between the orthocenter H and the corners. It is called the nine point circle or the Feuerbach circle.

With some effort, the proof can be seen from the above construction, using the facts in the construction of the Euler line.